Question

if =a 2+√3 then find the Value of a2 +1/a2​

Answer 1

EXPLANATION.

⇒ a = 2 + √3.

As we know that,

We can write equation as,

⇒ 1/a = 1/(2 + √3).

Now, rationalize the equation, we get.

⇒ 1/a = 1/(2 + √3) x (2 - √3)/(2 - √3).

⇒ 1/a = (2 - √3)/[(2)² - (√3)²].

⇒ 1/a = (2 - √3)/[4 - 3].

⇒ 1/a = (2 - √3)/1 = 2 - √3.

To find :

⇒ a² + 1/a².

⇒ a² = (2 + √3)².

⇒ a² = 4 + 3 + 4√3.

⇒ a² = 7 + 4√3.

⇒ 1/a² = (2 - √3)².

⇒ 1/a² = 4 + 3 - 4√3.

⇒ 1/a² = 7 - 4√3.

⇒ (a² + 1/a²) = [(7 + 4√3) + (7 - 4√3].

⇒ (a² + 1/a²) = [7 + 4√3 + 7 - 4√3].

⇒ (a² + 1/a²) = 14.

Answer 2

Correct Question :

• If a = 2+√3 then find the value of a² +1/a²

To Find :

• The value of a² + 1/a²

GivEn Data :

• a = 2 +√3

Identity Used :

$\green{ \star } \: \purple{ \underline{ \boxed{ \mathfrak{ {a}^{2} + \frac{1}{ {a}^{2}} = {(a + \frac{1}{a} )}^{2} - 2}}}} \: \red{\star}$

Concept :

Rationalisation :

• The process of multiplying a surd by another surd to get a rational number is called rationalisation.

• Each surd is called rationalising factor of another surd.

Rule to Rationalise the denominator of an expression :

• Multiply and divide the numerator and denominator of the given expression by rationalising factor of its denominator and simplify.

Solution :

It is given that the value of a is 2 + √3

So 1/a must be 1/(2 + √3) [By putting a's value in the denominator]

Now using the concept we get

$\sf \rarr \frac{1}{a}$

$\sf \rarr \frac{1}{2 + \sqrt{3} }$

$\sf \rarr\frac{1(2 - \sqrt{3} )}{(2 + \sqrt{3})(2 - \sqrt{3}) }$

Now as we can see that the denominator is (2 + √3)(2 - √3)

we need to apply identity to get the simplification easier i.e. :

(a + b)(a - b) = a² - b²

Here,

a ⟹ 2

b ⟹ √3

$\sf \rarr \frac{2 - \sqrt{3} }{ {(2)}^{2} - {( \sqrt{3} )}^{2} }$

$\sf \rarr \frac{2 - \sqrt{3} }{4 - 3}$

$\sf \rarr \frac{2 - \sqrt{3} }{1}$

$\underline{ \boxed{☢ \sf \: \frac{1}{a} = 2 - \sqrt{3} \: \: ☢ }}$

Now as per the given condition we need to find a² + 1/a²

$\leadsto \mathfrak{ {a}^{2} + \frac{1}{ {a}^{2} } = {(a + \frac{1}{a} )}^{2} - 2 }$

Putting the values we get

$\leadsto \mathfrak{ {a}^{2} + \frac{1}{ {a}^{2} } = {(2 + \sqrt{3} + 2 - \sqrt{3}) }^{2} - 2 }$

$\leadsto \mathfrak{ {a}^{2} + \frac{1}{ {a}^{2} } = {(2 + 2 + \cancel{ \sqrt{3} } - \cancel{ \sqrt{3} })}^{2} - 2 }$

$\leadsto \mathfrak{ {a}^{2} + \frac{1}{ {a}^{2} } = {(4)}^{2} - 2 }$

$\leadsto \mathfrak{ {a}^{2} + \frac{1}{ {a}^{2} } = 16 - 2 }$

$\star \: \blue{\underline{\boxed{ \green{\mathfrak{ {a}^{2} + \frac{1}{ {a}^{2} } = 14 }}}}}$

• Hence, the answer is 14

Learn More !!

$\star \: \underline{ \boxed{ \mathfrak{ {(a + b)}^{2} = {a}^{2} + 2ab + {b}^{2} }}}$

$\star \: \underline{ \boxed{ \mathfrak{ {(a - b)}^{2} = {a}^{2} - 2ab + {b}^{2} }}}$

$\star \: \underline{ \boxed{ \mathfrak{ {a}^{2} + {b}^{2} = {(a + b)}^{2} - 2ab }}}$

$\star \: \underline{ \boxed{ \mathfrak{ {a}^{2} + {b}^{2} = {(a - b)}^{2} + 2ab}}}$

$\star \: \underline{ \boxed{ \mathfrak{ {(a + b)}^{2} + {(a - b)}^{2} = 2( {a}^{2} + {b}^{2} )}}}$

$\star \: \underline{ \boxed{ \mathfrak{ {(a + b)}^{2} - {(a - b)}^{2} = 4ab }}}$

$\star \: \underline{ \boxed{ \mathfrak{ {(a + b + c)}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2(ab + bc + ca) }}}$

$\star \: \underline{ \boxed{ \mathfrak{ {(a + b - c)}^{2} = {a}^{2} + {b}^{2} + {c}^{2} + 2(ab + bc - ca) }}}$